Given Antisymmetric Matrix, $A$. Construct Matrix With Positive Real Part Eigenvalues of $A$
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Suppose you are given an antisymmetric matrix $A$, which is $2n times 2n$ and has entries which are complex.
Since $A$ is antisymmetric and is even dimensional, we know $n$ eigenvalues, call them $lambda_i$, have positive real part (suppose $A$ has no eigenvalue with real part $0$) and $n$ eigenvalues with negative real part.
The question is: What operations can I perform to $A$ so that it's eigenvalues are only the $lambda_i$?
(An example of such operations is I can shift the eigenvalues of $A$ by adding to $A$ some multiple of the identity matrix)
linear-algebra eigenvalues-eigenvectors
asked Aug 10 at 23:05
user1058860
283111
 |Â
show 7 more comments
up vote
-1
down vote
favorite
Suppose you are given an antisymmetric matrix $A$, which is $2n times 2n$ and has entries which are complex.
Since $A$ is antisymmetric and is even dimensional, we know $n$ eigenvalues, call them $lambda_i$, have positive real part (suppose $A$ has no eigenvalue with real part $0$) and $n$ eigenvalues with negative real part.
The question is: What operations can I perform to $A$ so that it's eigenvalues are only the $lambda_i$?
(An example of such operations is I can shift the eigenvalues of $A$ by adding to $A$ some multiple of the identity matrix)
linear-algebra eigenvalues-eigenvectors
asked Aug 10 at 23:05
user1058860
283111
Your question is a little unclear. What do you mean by projecting $A$ onto a subspace? Do you mean, find an $A$ which is a projection? Normally vectors are projected onto a subspace and this transformation can be represented by a matrix.
– Oliver Jones
Aug 11 at 20:50
Alright, I have edited the question. It should be clear now. I probably was misusing terminology.
– user1058860
Aug 11 at 20:58
Your question is still unclear. Are you asking: given a complex number $lambda$, find all antisymmetric matrices $A$ having $lambda$ as an eigenvalue? The condition Re$(lambda)ge 0$ is unnecessary.
– Oliver Jones
Aug 11 at 21:12
So I'm saying you're given an antisymmetric matrix, $A$. You know that for every eigenvalue of $A$, call it $lambda$, there exists another eigenvalue, $-lambda$. I would like to construct a matrix (not all) with $lambda$ as it's eigenvalues. My question is: can you construct such a matrix without knowledge of the eigenvectors and eigenvalues of $A$?
– user1058860
Aug 11 at 22:30
@user1058860 Your question is still completely unclear. I really can't understand what you're asking.
– Oliver Jones
Aug 11 at 23:38
 |Â
show 7 more comments
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Suppose you are given an antisymmetric matrix $A$, which is $2n times 2n$ and has entries which are complex.
Since $A$ is antisymmetric and is even dimensional, we know $n$ eigenvalues, call them $lambda_i$, have positive real part (suppose $A$ has no eigenvalue with real part $0$) and $n$ eigenvalues with negative real part.
The question is: What operations can I perform to $A$ so that it's eigenvalues are only the $lambda_i$?
(An example of such operations is I can shift the eigenvalues of $A$ by adding to $A$ some multiple of the identity matrix)
linear-algebra eigenvalues-eigenvectors
asked Aug 10 at 23:05
user1058860
283111
Suppose you are given an antisymmetric matrix $A$, which is $2n times 2n$ and has entries which are complex.
Since $A$ is antisymmetric and is even dimensional, we know $n$ eigenvalues, call them $lambda_i$, have positive real part (suppose $A$ has no eigenvalue with real part $0$) and $n$ eigenvalues with negative real part.
The question is: What operations can I perform to $A$ so that it's eigenvalues are only the $lambda_i$?
(An example of such operations is I can shift the eigenvalues of $A$ by adding to $A$ some multiple of the identity matrix)
linear-algebra eigenvalues-eigenvectors
asked Aug 10 at 23:05
user1058860
283111
edited Aug 12 at 1:27
asked Aug 10 at 23:05
user1058860
283111
asked Aug 10 at 23:05
user1058860
283111
asked Aug 10 at 23:05
user1058860
283111
283111
Your question is a little unclear. What do you mean by projecting $A$ onto a subspace? Do you mean, find an $A$ which is a projection? Normally vectors are projected onto a subspace and this transformation can be represented by a matrix.
– Oliver Jones
Aug 11 at 20:50
Alright, I have edited the question. It should be clear now. I probably was misusing terminology.
– user1058860
Aug 11 at 20:58
Your question is still unclear. Are you asking: given a complex number $lambda$, find all antisymmetric matrices $A$ having $lambda$ as an eigenvalue? The condition Re$(lambda)ge 0$ is unnecessary.
– Oliver Jones
Aug 11 at 21:12
So I'm saying you're given an antisymmetric matrix, $A$. You know that for every eigenvalue of $A$, call it $lambda$, there exists another eigenvalue, $-lambda$. I would like to construct a matrix (not all) with $lambda$ as it's eigenvalues. My question is: can you construct such a matrix without knowledge of the eigenvectors and eigenvalues of $A$?
– user1058860
Aug 11 at 22:30
@user1058860 Your question is still completely unclear. I really can't understand what you're asking.
– Oliver Jones
Aug 11 at 23:38
 |Â
show 7 more comments
Your question is a little unclear. What do you mean by projecting $A$ onto a subspace? Do you mean, find an $A$ which is a projection? Normally vectors are projected onto a subspace and this transformation can be represented by a matrix.
– Oliver Jones
Aug 11 at 20:50
Alright, I have edited the question. It should be clear now. I probably was misusing terminology.
– user1058860
Aug 11 at 20:58
Your question is still unclear. Are you asking: given a complex number $lambda$, find all antisymmetric matrices $A$ having $lambda$ as an eigenvalue? The condition Re$(lambda)ge 0$ is unnecessary.
– Oliver Jones
Aug 11 at 21:12
So I'm saying you're given an antisymmetric matrix, $A$. You know that for every eigenvalue of $A$, call it $lambda$, there exists another eigenvalue, $-lambda$. I would like to construct a matrix (not all) with $lambda$ as it's eigenvalues. My question is: can you construct such a matrix without knowledge of the eigenvectors and eigenvalues of $A$?
– user1058860
Aug 11 at 22:30
@user1058860 Your question is still completely unclear. I really can't understand what you're asking.
– Oliver Jones
Aug 11 at 23:38
Your question is a little unclear. What do you mean by projecting $A$ onto a subspace? Do you mean, find an $A$ which is a projection? Normally vectors are projected onto a subspace and this transformation can be represented by a matrix.
– Oliver Jones
Aug 11 at 20:50
Your question is a little unclear. What do you mean by projecting $A$ onto a subspace? Do you mean, find an $A$ which is a projection? Normally vectors are projected onto a subspace and this transformation can be represented by a matrix.
– Oliver Jones
Aug 11 at 20:50
Alright, I have edited the question. It should be clear now. I probably was misusing terminology.
– user1058860
Aug 11 at 20:58
Alright, I have edited the question. It should be clear now. I probably was misusing terminology.
– user1058860
Aug 11 at 20:58
Your question is still unclear. Are you asking: given a complex number $lambda$, find all antisymmetric matrices $A$ having $lambda$ as an eigenvalue? The condition Re$(lambda)ge 0$ is unnecessary.
– Oliver Jones
Aug 11 at 21:12
Your question is still unclear. Are you asking: given a complex number $lambda$, find all antisymmetric matrices $A$ having $lambda$ as an eigenvalue? The condition Re$(lambda)ge 0$ is unnecessary.
– Oliver Jones
Aug 11 at 21:12
So I'm saying you're given an antisymmetric matrix, $A$. You know that for every eigenvalue of $A$, call it $lambda$, there exists another eigenvalue, $-lambda$. I would like to construct a matrix (not all) with $lambda$ as it's eigenvalues. My question is: can you construct such a matrix without knowledge of the eigenvectors and eigenvalues of $A$?
– user1058860
Aug 11 at 22:30
So I'm saying you're given an antisymmetric matrix, $A$. You know that for every eigenvalue of $A$, call it $lambda$, there exists another eigenvalue, $-lambda$. I would like to construct a matrix (not all) with $lambda$ as it's eigenvalues. My question is: can you construct such a matrix without knowledge of the eigenvectors and eigenvalues of $A$?
– user1058860
Aug 11 at 22:30
@user1058860 Your question is still completely unclear. I really can't understand what you're asking.
– Oliver Jones
Aug 11 at 23:38
@user1058860 Your question is still completely unclear. I really can't understand what you're asking.
– Oliver Jones
Aug 11 at 23:38
 |Â
show 7 more comments
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Your question is a little unclear. What do you mean by projecting $A$ onto a subspace? Do you mean, find an $A$ which is a projection? Normally vectors are projected onto a subspace and this transformation can be represented by a matrix.
– Oliver Jones
Aug 11 at 20:50
Alright, I have edited the question. It should be clear now. I probably was misusing terminology.
– user1058860
Aug 11 at 20:58
Your question is still unclear. Are you asking: given a complex number $lambda$, find all antisymmetric matrices $A$ having $lambda$ as an eigenvalue? The condition Re$(lambda)ge 0$ is unnecessary.
– Oliver Jones
Aug 11 at 21:12
So I'm saying you're given an antisymmetric matrix, $A$. You know that for every eigenvalue of $A$, call it $lambda$, there exists another eigenvalue, $-lambda$. I would like to construct a matrix (not all) with $lambda$ as it's eigenvalues. My question is: can you construct such a matrix without knowledge of the eigenvectors and eigenvalues of $A$?
– user1058860
Aug 11 at 22:30
@user1058860 Your question is still completely unclear. I really can't understand what you're asking.
– Oliver Jones
Aug 11 at 23:38